Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a geometric construction using the ancient Greek rules, it must be either rational or a very special kind of algebraic number known as a Euclidean number. Because the number is transcendental, the construction cannot be done according to the Greek rules.

Liouville showed how to construct special cases (such as Liouville's constant) using Liouville's approximation theorem. In particular, he showed that any number that has a rapidly converging sequence of rational approximations must be transcendental. For many years, it was only known how to determine if special classes of numbers were transcendental. The determination of the status of more general numbers was considered an important enough unsolved problem that it was one of Hilbert's problems.

Great progress was subsequently made by Gelfond's theorem, which gives a general rule for determining if special cases of numbers of the form are transcendental. Baker produced a further revolution by proving the transcendence of sums of numbers of the form for algebraic numbers and .